The above computation is purely quantum group representation-theoretic, and its keys are only (REF ) and the branching rule for \(U_q\mathfrak {gl}(m)\otimes U_q\mathfrak {gl}(n) \hookrightarrow U_q\mathfrak {gl}(m+n)\) .
The actual computation for the classification of irreducible spherical unitary representations of \(\mathrm {U}_q(\infty ) < \mathrm {U}_q(\infty )\times \mathrm {U}_q(\infty )\)  comes down to the analysis on the link (REF ) due to Gorin [1]}.
